Quantum pattern matching fast on average

The d-dimensional pattern matching problem is to find an occurrence of a pattern of length m × dots × m within a text of length n × dots × n, with n ≥ m. This task models various problems in text and image processing, among other application areas. This work describes a quantum algorithm which solves the pattern matching problem for random patterns and texts in time widetildeO((n/m)d/2 2^O(d3/2√log m)). For large m this is super-polynomially faster than the best possible classical algorithm, which requires time widetildeOmega( (n/m)d + nd/2 ). The algorithm is based on the use of a quantum subroutine for finding hidden shifts in d dimensions, which is a variant of algorithms proposed by Kuperberg.